PomeF8F49090AB94F8F49090AB94
C7AB6352651BHere we recall the origin of various results.
section 5.7.
The notion of plurisubharmonic functions was introduced by Lelong [Lel45] and Oka [Oka42], based on F. Riesz's theory of subharmonic functions [Rie26]. See [Bre72] for an excellent introduction to the early history of the subject. We refer to [Bre65] for the foundations of potential theory and [GZ17] for the pluripotential theory.
The study of Josefson-type theorems was initiated by Josefson [Jos78], who proved theorem 5.13.19. The global Josefson theorem theorem 5.13.20 was due to Vu [Vu19]. In the projective setting, it was due to Dinh–Sibony [DS06] and in the Kähler setting, it was established by Guedj–Zeriahi [GZ05].
The extension theorem theorem 5.14.10 was proved in [GR56]. In fact, they proved a more general version for complex spaces, see theorem 9.10.14. For some related important extension theorems, see [Shi72, Wan22, CGZ13].
proposition 5.14.15 was due to Kiselman [Kis78].
The idea of fine topology was due to H. Cartan [Car41]. The plurifine topology was introduced by Fuglede during the Séminaire d'analyse de Lelong–Dolbeault–Skoda of the year 1983/1984 [Lel86], but often wrongly attributed to Bedford–Taylor. The key result theorem 5.15.11 was claimed in Bedford–Taylor's work [Theorem 2.3, BT87] without proof. The first rigorous proof was given by El Marzguioui–Wiegerinck [EMW06]. A weaker result was proved earlier in [Theorem 4.8.7, Kl91].
Results in 5.15.17 are certainly well-known and are already implicitly used in the literature. I could not find the proofs in the literature, so all details are presented.
The semicontinuity theorem theorem 5.16.7 was due to Siu [Siu74].
Multiplier ideal sheaves were introduced by Nadel [Nad90]. The idea of theorem 5.16.18 first appeared in the ground-breaking work of Boucksom–Favre–Jonsson [BFJ08].
The openness conjecture was formulated and studied by Demailly–Kollár [DK01] and later proved by Berndtsson [Bern13b]. The strong openness theorem 5.16.21 was first established by Guan–Zhou [GZ15].
5.18.15 was due to [Proposition 4.1.6, Dem15].
The notions of positivity of currents definition 5.19.1 are due to Lelong [Lel68] and Harvey–Knapp [HK74], although it is often mistakenly attributed to Demailly.
The $L^2$ extension theorems such as theorem 5.16.26 were first studied by Ohsawa–Takegoshi [OT87]. Since then, it has become a very active area. It is almost impossible to give a complete list of the various generalizations and improvements.
section 13.1
The Monge–Ampère operators for bounded plurisubharmonic functions were introduced by Bedford–Taylor [BT76, BT82]. The non-pluripolar product was due to Bedford–Taylor [BT87], Guedj–Zeriahi [GZ07] and Boucksom–Eyssidieux–Guedj–Zeriahi [BEGZ10].
example 13.11.8 was claimed in [BEGZ10] right after Definition 1.1 without proof. The first proof was essentially due to El Kadiri in [EK24].
The semicontinuity theorem theorem 13.13.9 and the domination principle theorem 13.13.43 were due to [DDNL18fullmass]. The monotonicity theorem was due to [WN19, DDNL18mono].
The key lemma 13.13.30 was proved in [DDNLmetric]. theorem 13.13.24 was due to [DDNLsurv].
section 16.1
16.3.3 was proved in [Lemma 3.7, DDNL18mono].
The notion of the $P$-envelope is due to Ross–Witt Nyström [RWN14] based on the ideas of Rashkovskii–Sigurdsson [RS05].
theorem 16.3.12 is due to [Theorem 3.8, DDNL18mono]. The diamond inequality theorem 16.3.35 and proposition 16.3.25 are due to [Theorem 5.4, DDNLmetric]. Most results in 16.5 are simple generalizations of the corresponding results in [DDNL18big, DDNL18fullmass].
The study of finite energy potentials was initiated by Cegrell [Ceg98] in the local setup; see also his general definition of the complex Monge–Ampère operator [Ceg04]. The global setup was first studied by Guedj–Zeriahi [GZ07]. The variational approach to degenerate complex Monge–Ampère equations in big cohomology classes was developed by Boucksom–Eyssidieux–Guedj–Zeriahi [BEGZ10] and Berman–Boucksom–Guedj–Zeriahi [BBGZ13]. For the complex Monge–Ampère equation and pluripotential methods for weak solutions, see Kołodziej [Kol98, Kol05] and the references therein.
The $\mathcal{I}$-envelope was introduced by Darvas–Xia [DX22], inspired by the works of Dano Kim [Kim15] and Boucksom–Favre–Jonsson [BFJ08]. The notion of $\mathcal{I}$-model singularities was first formulated in the explicit way in [DX22] in 2020, although it was already essentially known in Boucksom–Jonsson's work. In fact, they correspond exactly to the homogeneous non-Archimedean potentials assuming that the relevant masses do not vanish. A less explicit equivalent formulation of $\mathcal{I}$-model potentials also appeared in [Dem15]. A few months later, the same notion was rediscovered by Trusiani [Tru22].
theorem 13.13.57 is due to [DNT21].
section 17.10
The study of geodesics in the space of Kähler metrics was initiated by Mabuchi [Mab87], Semmes [Sem92] and Donaldson [Don99]. The existence of geodesics with smooth boundary data was established by Chen [Chen00].
The metric geometry and Mabuchi completion of finite-energy classes in the Kähler case were developed by Darvas [Dar15, Dar17]. The notion of weak geodesics was studied in detail by Darvas [Da17] in the Kähler case.
The case of general big classes was partly handled in [DDNL18fullmass], [DDNL18big]. However, the key fact that the geodesics between two full mass potentials have the correct limit at the end points does not seem to have been proved in any references. We give a proof in proposition 17.10.24. We also extend the relevant results to the relative setting.
In the special case $\phi =V_\theta $, proposition 17.10.65 was proved in [Theorem 3.12, DDNL18fullmass], while the sequential case of proposition 17.10.77 was proved in [Proposition 3.3, DDNL18fullmass].
Previously, proposition 17.10.29 and proposition 17.10.39 were only known in the Kähler case.
Most results in subsection 17.10.40 are simple extensions of [DDNL18big].
section 21.12
Classical toric Kähler geometry is rooted in the works of Atiyah [Ati82], Delzant [Del88], Guillemin [Gui94], Abreu [Abr98] and Donaldson [Don02]. The pluripotential-theoretic toric framework used here was first written down by Berman–Berndtsson [BB13] and Coman–Guedj–Sahin–Zeriahi [CGSZ19].
The beautiful theorem theorem 21.12.31 was first proved by Yi Yao, who did not publish the result. Later on, a new proof was found by Botero–Burgos Gil–Holmes–de Jong [BBGHdJ21]. We chose to present the approach of Yao, which integrates naturally with our framework.
Proposition 21.12.39 is an extension of [Proposition 2.9, BB13].
section 16.7
The notion of $P$-preorder is new, as well as most results in section 16.8.
The $d_ S$-pseudometric was introduced in [DDNLmetric]. The basic properties are proved in [DDNLmetric] and [Xia21].
example 16.8.19 was due to Berman–Boucksom–Jonsson [BBJ21].
theorem 16.9.45 can be viewed as a vast generalization of [Cao15], where a question raised by Demailly was confirmed.
theorem 16.9.55 is proved in [Xia22]. theorem 16.9.59 and theorem 16.9.57 appear to be new. These results appeared previously in the form of lecture notes.
section 18.1
The notion of $\mathcal{I}$-good singularities was due to [DX21]. The name $\mathcal{I}$-good was chosen in [Xia22].
The key results theorem 18.1.2 and theorem 18.1.46 are due to [DX21, DX22]. Previously, Rashkovskii [Ra13] also studied the local approximation problem with a different motivation.
There are some further examples of $\mathcal{I}$-good singularities provided by [BBGHdJ21] with applications in the theory of modular forms in [BBGHdJ22].
section 20.2
The trace operator was introduced in [DX24]. Here we present a different point of view. theorem 20.2.39 was proved in [DX24].
The analytic Bertini theorem theorem 20.2.45 was proved in [XiaBer], based on the works of Matsumura–Fujino [FM21] and Fujino [Fuj23]. A weaker result was established by Meng–Zhou [MZ23].
section 17.11
The technique of test curves originates from [RWN14]. It was generalized by Darvas–Di Nezza–Lu [DDNL18big], [DX21], [DZ22] and [DXZ23]. We give the full details of the proofs.
Test curves in definition 17.12.1 are called maximal test curves in the literature, a terminology which I do not like. I prefer to call the usual notion of test curves in the literature sub-test curves.
proposition 17.13.35 was first proved by He–Testorf–Wang in [HTW23]. proposition 17.13.38 was due to Hisamoto [His16].
17.14.15 was a folklore result. I am unaware of any written proof in the literature before our paper [DX22]. Finski [Fin22b] also gave a different proof with different techniques later on.
definition 17.14.7 was not the original definition of maximal geodesic rays of Berman–Boucksom–Jonsson in [BBJ21]. One of the first major applications of our theory was this pluripotential-theoretical characterization of maximal geodesic rays, as proved in our very first paper [DX22].
example 17.14.10 is essentially due to Ross–Witt Nyström [RWN14]. The Phong–Sturm geodesics were first constructed in [PS07]. For the related link between test configurations, filtrations and Okounkov bodies, see Boucksom–Chen [BC11] and Witt Nyström [WN12]. For filtrations and generalized test configurations in K-stability, see Székelyhidi [Sze15].
Results in section 17.15 are easy generalizations of the results proved in [Xia23Operations].
section 19.1
The algebraic theory of partial Okounkov bodies was developed in [Xia21]. The transcendental Okounkov body was first defined by Deng [Deng17] as suggested by Demailly. The volume identity was proved in [DRWNXZ]. The transcendental theory of partial Okounkov bodies is new.
section 20.1
The notion of b-divisors was introduced by Shokurov [Sho93] in the context of minimal model programs.
The applications of b-divisors in pluripotential theory began with [BFJ09]. The intersection theory of nef b-divisors was introduced by Dang–Favre [DF20]. The technique of singularity b-divisors was introduced in [XiaPPT] in 2020. The general form first appeared in [Xia22]. One year later, a special case was rediscovered in [BBGHdJ21].
The current chapter reproduces a large part of [Xiabdiv]. For further developments, we refer to [Xiabdiv2].
example 20.1.9 is reproduced from [Example 3.3, DF20].
section 21.13
The whole chapter appears to be new. The study of toric pluripotential theory on big line bundles was made possible by the development of partial Okounkov bodies. The key result is theorem 21.13.45.
Most results in this chapter resulted from discussions with Yi Yao.
section 22.1
Most results in this chapter come from [Xia23Operations]. Results from subsection 22.1.54 are new, although the main idea was already contained in [Xia21]. The terminology of Duistermaat–Heckman measures goes back to the classical theorem of Duistermaat–Heckman [DH82]; see also their addendum [DH83]. In the setting of Kähler geometry, the Duistermaat–Heckman measures were studied in detail by Boucksom–Hisamoto–Jonsson [BHJ17].
Foundational non-Archimedean global pluripotential theory was developed by Boucksom–Favre–Jonsson [BFJ16b] and, over a trivially valued field, by Boucksom–Jonsson [BJ22].
theorem 22.1.111 is due to [DXZ23]. An alternative approach to the transcendental theory is due to Mesquita-Piccione [MP24]. For further progress in this direction, we refer to [MPWN25, WN25].
Special cases of the results in this section have been applied to study K-stability, see [XiaPPT], [DZ22], [DXZ23] and [DR22]. In [DX22], we established the bijective correspondence between a class of $\mathcal{I}$-model test curves and the maximal geodesic rays in the sense of [BBJ21].
section 18.28
The study of asymptotic expansions of Bergman kernels in Kähler geometry was initiated by Tian [Ti90] with later improvements due to Bouché [Bou90], Ruan [Rua98], Catlin [Cat99], Zelditch [Zel98], Lu [Lu00] among many others.
The special case of theorem 18.28.49 without the prescribed singularity $\varphi $ was due to Berman–Boucksom–Witt Nyström, see [BB10], [BBWN11]. The general case is due to [DX21].
8B52D455AC39We give a list of important open problems in this theory.
We do not repeat the conjectures mentioned in the main text.
Conjecture 25.23.3.F96A00F2C17A Let $X$ be a connected compact Kähler manifold and $Y$ be a submanifold. Fix a Kähler class $\alpha $ on $X$. For each Kähler current $S\in \alpha |_ Y$, we can find a Kähler current $T\in \alpha $ such that
Problem 25.23.4.6DBA7A1F707B Is it possible to extend the definition of the trace operator $\operatorname{Tr}_ Y$ to the case where the ambient variety is only unibranch?
Problem 25.23.5.6BD8E77C1AAF Is there a natural definition of the transcendental Okounkov body of a closed positive $(1,1)$-current $T$ with $0$-mass so that its dimension is equal to the numerical dimension of $T$?
Problem 25.23.6.0C91C5C2794A Consider a compact Kähler manifold $X$ and a connected submanifold $Y$. We have defined the trace operator $\operatorname{Tr}_ Y$ from a subset of $\mathrm{QPSH}(X)/\sim _\mathcal {I}$ to $\mathrm{QPSH}(Y)/\sim _\mathcal {I}$. Is it possible to refine this operator to one from a subset of $\mathrm{QPSH}(X)/\sim _ P$ to $\mathrm{QPSH}(Y)/\sim _ P$?
Problem 25.23.7.7FAD0FAF17B7 Consider a connected compact Kähler manifold $X$ of dimension $n$ and a smooth flag $Y_{\bullet }$ on $X$. Consider a closed smooth real $(1,1)$-form $\theta $ on $X$ representing a big cohomology class and $\varphi \in \mathrm{PSH}(X,\theta )$ with $\int _ X \theta _{\varphi }^ n>0$.
Can one define a refined notion of partial Okounkov bodies $\Delta '_{Y_{\bullet }}(\theta +\mathrm{dd}^{\mathrm{c}}\varphi )$ contained in $\Delta _{Y_{\bullet }}(\theta +\mathrm{dd}^{\mathrm{c}}\varphi )$ with volume given by $\frac{1}{n!}\int _ X \theta _{\varphi }^ n$?
Problem 25.23.8.856DCCA91F34 To what extent can the results in the current book be generalized to the non-Kähler setting?
The non-pluripolar products in the non-Kähler setting were recently studied by Boucksom–Guedj–Lu in [BGL24]. See also the references therein.
Problem 25.23.9.18561C2CB4F4 To what extent can the results in the current book about closed positive $(1,1)$-currents be generalized to closed positive currents of higher bidegree?
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